Geometrically nonlinear analytic solution for circular liquid inclusions in a soft elastic solid

We re-examine the plane deformation of a compressible circular liquid inclusion embedded within an elastic, saturated solid matrix under uniform far-field loading. In contrast to the classical solution for this problem which predicts only a linear elastic response of the composite system to the far-field loading, we identify a modified but still closed-form solution allowing for a nonlinear response to the far-field loading. The modified solution differs from its classical counterpart mainly in that it additionally captures the directional change of the liquid pressure when the liquid-solid interface is deformed with the far-field loading. In this case, the modified solution offers a possibility of characterizing, to some extent, the geometrically nonlinear behavior of a soft elastic solid filled with liquid inclusions under relatively large external loadings. Numerical examples are presented to demonstrate the essential improvements brought by the modified solution as opposed to the classical solution in predicting the local stress field and the overall effective moduli of the homogenized liquid-solid composite materials. Since the overall framework of the modified solution is still confined to linear elasticity, however, it inevitably has some limitations in applications: it works well only for a weakly nonlinear soft matrix under moderate far-field tensile strain (for example, up to around 14 %), although it fails basically for a soft matrix under compression loadings.